Modeling of underwater vehicle s dynamics

Transcription

1 Proceeding of the 11th WEA International Conference on YTEM, Agio Nikolao, Crete Iland, Greece, July 23-25, Modeling of underwater vehicle dynamic ANDRZEJ ZAK Department of Radiolocation and Hydrolocation Naval Univerity of Gdynia midowicza 69, Gdynia POLAND Abtract: - The paper preent algorithm of underwater vehicle dynamic modeling uing technique of artificial neural network. Paper include mathematical decription of ued dynamical neuron and bai of it learning proce. Next the reult of reearch for real underwater vehicle were preented. Key-Word: - artificial neural network, dynamic modeling, underwater vehicle 1 Introduction Modeling in control theory mean the proce in which reult, baing on input and output ignal, arie the mathematical model of object admitted a the bet according to the accepted criterion. By the object model it i undertood the preentation of intereted, eential propertie in convenient form. Therefore procedure of modeling can be, in the implet way, preented a it i done on figure 1 [2]. Fig. 1. chematic preentation of modeling proce Model appear in many field of human knowledge. In all cae poeion of model allow for preliminary teting of creating ytem what permit to horten the time neceary for projection phae and inculcate of ytem. It alo prevent from accidental damage of real object. Often modeling tart from apply ome phyical law which take part in invetigated proce. But it i only poible when thoe phenomenon are quite imple. If the numerical value of all external and internal condition of modeled object are known and phyical knowledge about thi object i full it i poible to calculate value of all parameter. However thoe cae are rare becaue of knowledge hortage about internal proce and indefinablene carried in by environment. In other hand very often they are only intereting dependencie between object input and output ignal accepted that input ignal are control ignal and meaured output ignal give information about object tate. In thi cae object can be treated a ingle or multi input and ingle or multi output object with unknown tructure. In thi cae more often the method which can adjut the etablihed tructure bai on the experimental data like artificial neural network are ued. Thi technique, with regard on it unquetionable advantage, i ued a effective method in modeling of object dynamic. To the mot important feature of neural network can be included: - the poibility of approximation of any non-linear repreentation; - the parallel proceing of information; - the poibility of learning and adaptation; - ignal proceing from many input and generating of many output (multidimenional ytem). Therefore the paper preent technique of neural network ued to dynamic identification of underwater vehicle. 2 Neural algorithm of dynamic identification 2.1 Dynamical neural network Becaue the procee, taking place during movement of underwater vehicle, have dynamic character, neuronal modeling in uch cae require pecial olution. One of the poibilitie i to ue the recurrent neural network a Hopfield network or the implet backpropagation neural network with recurrent connection. In the lat cae the effect of dynamic behavior i obtain by widening input vector with the information of model output in previou moment. In lat invetigation in aim of obtainment of dynamic character of neural network the dynamic become introduced to the neuron in uch way that neuron activation depend on it internal tate [1]. It i done by

2 Proceeding of the 11th WEA International Conference on YTEM, Agio Nikolao, Crete Iland, Greece, July 23-25, introducing a linear dynamic ytem to the neuron tructure. In thi way each neuron in dynamic network reproduce the pat ignal value with two ignal: the input ignal and output ignal. Figure 2 repreent the tructure of uch neuron with one output and many input. Fig. 2. The model of dynamic neuron Three main operation are performed in thi dynamic tructure. Firt of all, in block of ummation, the weighted um of input i calculated according to the formula: N ϕ = w i x i (1) i= 1 where: w input weight vector; x input vector. Calculated weighted um i proceed in filter block which can be the linear dynamic ytem of any order. Thi ytem conit of delay element and feedback and feed-forward path weighted by the vector weight T T a = [ a1, a2,, a n ] and b = [ b0, b1,, b n ], repectively a it i hown at figure 3. Thi tructure can be decribed by the following difference equation: γ = a1γ 1) anγ n) + (2) + b0ϕ( + b1ϕ 1) + + bnϕ( k n) where: ϕ input of filter block; γ output of filter block; a, b the vector weight of feedback and feedforward path. where: F non-linear activation function; parameter of the activation function. g lope 2.2 Learning proce The main objective of learning algorithm i to adjut all unknown parameter of dynamic neuron uch a input weight w, value of filter parameter a and b, and lope parameter of activation function g, baed on the given data et of input-output pair [3, 4, 6]. In fact, the training proce involve the determination of unknown network parameter that minimize a performance index J baed on an error function defined a: 1 J = E{ e( 2 } (4) 2 where: E denote the mean value operator e = d( y( ; d ( - deired repone of neural network, y ( - actual repone of neural network. To olve the minimization problem (4) and find optimal network parameter the conjugate gradient algorithm may be applied. Aumption that correction of network parameter i done after each learning pattern according to the formula: v ( l+ 1) = v( l) η J + µ J (5) v v= v( l) v v= v( l 1) where: v w, a, b, g ) - the generalized network ( parameter, η - learning rate, µ - coefficient of conjugate, l - iteration index. Taking into account the performance index (4) the gradient i given by: J = E{ e F ( g γ ) v } (6) v where: v ( - enitive vector of g γ ( to the change o parameter v, F ( g γ ) - derivative of the activation function with repect to v. According to the above formulae can be defined the individual element of the enitive vector v ( for the element of unknown generalized parameter v w, a, b, g ) ( Fig. 3. Dynamic linear ytem Finally, the neuron output which i output ignal of activation block can be decribed a: y = F[ g γ ] (3)

3 Proceeding of the 11th WEA International Conference on YTEM, Agio Nikolao, Crete Iland, Greece, July 23-25, wi a p bq g = g = g = g for = g for n = w p= 1,..., n 0, 1,..., g = g n bq xi q) a p w p) i p = g γ p) a q= 0 = 1 q= = g ϕ( k q) b i q p n = γ 2.3 Algorithm of identification Algorithm of underwater vehicle dynamic identification uing neural technique preented above and ued in reearch can be preented a it wa done on figure 4 [7, 8]. At firt phae of neural algorithm the network i initialized o the following tep are executed: - generating the neuron with tructure decribed by (3) - etup the beginning value of network parameter like: input weight, coefficient of weight of filter block path, lope parameter; - zero out the other parameter; - etup the connection between neuron. The activation function accepted in reearch i tangenoidal function decribed by equitation: 1 y = (8) ( x( k ) 1+ e The number of neuron in input layer of neural network i determined by control vector ize, and number of neuron in output layer of network in equal to ize of oberved tate vector. The hidden layer configuration wa pecified uing Vapnik-Chervonenki dimenion [6]. The connection between neuron were made each other to each other between neighboring layer without recurrent connection and connection which kip any layer becaue of preented neural network feature. The econd phae of identification algorithm i taking the control vector, it normalization, and next calculation the repone of neural network. Next average quare error i calculated baed on network repone and meaured tate vector of vehicle. Thi parameter mean about adjutment of neural model to the real object. If network i in reearch phae of identification the parameter are changed according to backpropagation (7) method [6] baing on value of average quare error and method of conjugate gradient preented above. If the algorithm i not finihed by uer or after ucceed the accepted level of average quare error the proce begun from taking next control vector. tart Initialization of neural network Taking the control vector Calculating the repone of neural network Taking the tate vector Calculation of average quare error I thi learning phae? Actualization of neural network parameter Viualiation and aving of neural network inputand output I thi end of reearch? top Ye Ye Fig. 4. Algorithm of neural modeling of underwater vehicle 3 Reult of reearch Remotely operated, underwater vehicle UKWIAL which wa deigned in Poland by team from Univerity of Gdank wa the object of reearch. Baic technical parameter of thi vehicle are: - working depth: 200 m; - vehicle ma: 175 kg; No No

4 Proceeding of the 11th WEA International Conference on YTEM, Agio Nikolao, Crete Iland, Greece, July 23-25, equipment: - two TV camera; - weeping onar; - onic finder; - magnetic and electronic compa; - electrolytic inclination gauge; - quantity of thruter: 6, 4 in horizontal and 2 in perpendicular plane; - force of individual thruter: 220 N; - maximum forward peed: 1,5 m /. The appearance of object of reearch i preented on figure 5. Fig. 5. Object of reearch - remotely operated, underwater vehicle Ukwial The reearch proce wa dividend into two phae. In firt phae, called identification, the parameter of created model were changed due to preented above algorithm. In econd phae, called imulation, the parameter of model were not changed and only the error of repone of object and created neural model wa upervied. The control vector during reearche ha form: u = uτ, τ, τ ] (9) [ X Z N where: τ X - force along longitudinal axi of vehicle; τ Z - force along tranvere axi of vehicle; τ N - rotatory moment around normal axi of vehicle. The value for all component of control vector were etup by inclination of two joytick. The tate vector which can be oberved during reearche ha form: x= x[ d, ψ ] (10) where: d - depth of immerion with preciion to 0.1 [m]; ψ - bearing angle of vehicle with preciion to 0.5 o. The reult of reearche were preented on figure below. The light line how value of component of real object control vector or tate vector, the dark line how value calculated by neural model. Fig. 6. The value of component of control vector in identification phae

5 Proceeding of the 11th WEA International Conference on YTEM, Agio Nikolao, Crete Iland, Greece, July 23-25, Fig. 7. The value of component of tate vector in identification phae Fig. 8. The um quare error of repone of object and neural model in identification phae Fig. 9. The value of component of control vector in imulation phae

6 Proceeding of the 11th WEA International Conference on YTEM, Agio Nikolao, Crete Iland, Greece, July 23-25, between repone of object and neural model mean about it very good preciion neceary during another feature reearch. The feature reearch will concentrated on uing the elforganization method for neural network ued for modeling dynamic of underwater vehicle. More over the other method of learning neural network hould be checked at purpoe to peed up the learning proce. Another point i to ue thi algorithm for another, multidimenional object. Created neural model can be ued a platform for imulation during projection phae of building control ytem and alo a bai of imulator for training of pilotage of underwater vehicle. Fig. 10. The value of component of tate vector in imulation phae Reference: [1] Back A. D., Toi A. C., FIR and IIR ynape, A new neural network architecture for time erie modeling, Neural Computation 3, pp , 1991 [2] Gutenbaum J., Mathematical ytem Modeling,. Academical Printing Houe Exit, Waraw 2003 [3] Konińki R. A., Artificial Neural Network, Nonlinear dynamic and chao, WNT, Waraw 2002 [4] Duch W., Korbicz J., Rutkowki L., Tadeuiewicz R., Neural network, Academical Printing Houe Exit, Waraw 2004 [5] Foen Thor I., Guidance and control of ocean vehicle, John Wiley & on, Chicheter 1994 [6] Oowki.: Neural network in algorithm conception, Printing Houe of Technical Academy of Waraw, Waraw 1996 [7] Żak A., Dynamic identification of remotely operated vehicle in exploational condition, Naval Univerity of Gdynia, Gdynia 2006 [8] Żak A., Neural identification of underwater vehicle dynamic, XV International Conference of Automation, Waraw, pp , 2005 Fig. 11. The um quare error of repone of object and neural model in imulation phae 4 Concluion A it i hown at reult of reearche the preented neural network technique i ueful for modeling the dynamic of underwater vehicle. Mean value of average quare error in phae of identification during reearch wa about 0.03 and in imulation phae it take about It mean that accepted method of learning and the configuration of neural network ued a vehicle dynamic neural model wa properly choen. Moreover the mall different